
/* @(#)e_exp.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *	Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *	accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *	the interval [0,0.34658]:
 *	Write
 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Reme algorithm on [0,0.34658] to generate
 * 	a polynomial of degree 5 to approximate R. The maximum error
 *	of this polynomial approximation is bounded by 2**-59. In
 *	other words,
 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *  	(where z=r*r, and the values of P1 to P5 are listed below)
 *	and
 *	    |                  5          |     -59
 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *	    |                             |
 *	The computation of exp(r) thus becomes
 *                             2*r
 *		exp(r) = 1 + -------
 *		              R - r
 *                                 r*R1(r)
 *		       = 1 + r + ----------- (for better accuracy)
 *		                  2 - R1(r)
 *	where
 *			         2       4             10
 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *	From step 1, we have
 *	   exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *	exp(INF) is INF, exp(NaN) is NaN;
 *	exp(-INF) is 0, and
 *	for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *	according to an error analysis, the error is always less than
 *	1 ulp (unit in the last place).
 *
 * Misc. info.
 *	For IEEE double
 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include "fdlibm.h"
#include "math_config.h"
#if __OBSOLETE_MATH_DOUBLE

#ifdef _NEED_FLOAT64

static const __float64
    one         = _F_64(1.0),
    halF[2]	= {_F_64(0.5),_F_64(-0.5),},
    huge	= _F_64(1.0e+300),
    twom1000    = _F_64(9.33263618503218878990e-302),     /* 2**-1000=0x01700000,0*/
    o_threshold = _F_64(7.09782712893383973096e+02),  /* 0x40862E42, 0xFEFA39EF */
    u_threshold = _F_64(-7.45133219101941108420e+02),  /* 0xc0874910, 0xD52D3051 */
    ln2HI[2]    ={ _F_64(6.93147180369123816490e-01),  /* 0x3fe62e42, 0xfee00000 */
    _F_64(-6.93147180369123816490e-01),},/* 0xbfe62e42, 0xfee00000 */
    ln2LO[2]    ={ _F_64(1.90821492927058770002e-10),  /* 0x3dea39ef, 0x35793c76 */
    _F_64(-1.90821492927058770002e-10),},/* 0xbdea39ef, 0x35793c76 */
    invln2      = _F_64(1.44269504088896338700e+00), /* 0x3ff71547, 0x652b82fe */
    P1          = _F_64(1.66666666666666019037e-01), /* 0x3FC55555, 0x5555553E */
    P2          = _F_64(-2.77777777770155933842e-03), /* 0xBF66C16C, 0x16BEBD93 */
    P3          = _F_64(6.61375632143793436117e-05), /* 0x3F11566A, 0xAF25DE2C */
    P4          = _F_64(-1.65339022054652515390e-06), /* 0xBEBBBD41, 0xC5D26BF1 */
    P5          = _F_64(4.13813679705723846039e-08); /* 0x3E663769, 0x72BEA4D0 */

__float64
exp64(__float64 x) /* default IEEE double exp */
{
    __float64 y, hi, lo, c, t;
    __int32_t k = 0, xsb;
    __uint32_t hx;

    GET_HIGH_WORD(hx, x);
    xsb = (hx >> 31) & 1; /* sign bit of x */
    hx &= 0x7fffffff; /* high word of |x| */

    /* filter out non-finite argument */
    if (hx >= 0x40862E42) { /* if |x|>=709.78... */
        if (hx >= 0x7ff00000) {
            __uint32_t lx;
            GET_LOW_WORD(lx, x);
            if (((hx & 0xfffff) | lx) != 0)
                return x + x; /* NaN */
            else
                return (xsb == 0) ? x : _F_64(0.0); /* exp(+-inf)={inf,0} */
        }
        if (x > o_threshold)
            return __math_oflow(0); /* overflow */
        if (x < u_threshold)
            return __math_uflow(0); /* underflow */
    }

    /* argument reduction */
    if (hx > 0x3fd62e42) { /* if  |x| > 0.5 ln2 */
        if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
            hi = x - ln2HI[xsb];
            lo = ln2LO[xsb];
            k = 1 - xsb - xsb;
        } else {
            k = invln2 * x + halF[xsb];
            t = k;
            hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
            lo = t * ln2LO[0];
        }
        x = hi - lo;
    } else if (hx < 0x3df00000) { /* when |x|<2**-32 */
        if (huge + x > one)
            return one + x; /* trigger inexact */
    }

    /* x is now in primary range */
    t = x * x;
    c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
    if (k == 0)
        return one - ((x * c) / (c - _F_64(2.0)) - x);
    else
        y = one - ((lo - (x * c) / (_F_64(2.0) - c)) - hi);
    if (k >= -1021) {
        __uint32_t hy;
        GET_HIGH_WORD(hy, y);
        SET_HIGH_WORD(y, hy + lsl(k, 20)); /* add k to y's exponent */
        return y;
    } else {
        __uint32_t hy;
        GET_HIGH_WORD(hy, y);
        SET_HIGH_WORD(y, hy + lsl((k + 1000), 20)); /* add k to y's exponent */
        return y * twom1000;
    }
}

_MATH_ALIAS_d_d(exp)

#endif /* _NEED_FLOAT64 */
#else
#include "../common/exp.c"
#endif /* __OBSOLETE_MATH_DOUBLE */
